In one form or another, the bulk of fractal designs seen in art are created by the perpetuation of a mathematical anomaly.  This perpetuation does not always have to occur based on a standard IFS.  For all intents and purposes, an IFS is just one “variation”, or method of fractal control.  Some fractal programs refer to it as an IFS, while others call it a “Barnsley” (find more information on "IFS" and a spotlight on “Barnsley’s Fern” in the section on Iterated Function Systems).  Many other variations exist to manipulate fractals in 2D and 3D space.  Some of these types, such a “hyperbolic” variation, are generally only used to morph an existing design, while others, like the “julia”, can also be used as the primary method of perpetuation.  Listed below are some quick descriptions of popular fractal variations and their effects, included here primarily to open the mind to the unending possibilities of fractal scripting.  That being said, this list is only a small handful of the variations that currently exist in the fractal design world.  The best way to learn about these and other variations is through practice and experimentation.

JulianJulian Example
Julia fractals are some of the most complicated fractals to describe, yet also among the most visually interesting.  A julia set is the result of a self-perpetuating quadratic polynomial.  More specifically, it is an equation involving two variables.  The first variable is a constant number or expression, and the second variable continuously changes itself by adding the first variable to its own square.  Sound chaotic?  Well, of course it is.  It’s a fractal!  The end result of this complicated affair is a fractal that resembles an IFS in some ways, but its iterations are developed from relative points in space rather than from the points of a solid base shape.  That is to say, instead of continuing out from itself in a single direction creating a fern-like pattern, it grows outward in multiple directions at specific intervals like something of a fractal explosion.  This method is known as “formula iteration”.

Julia fractals are generally easy to spot, as the nature of this variation often forms multiple lines of geometric symmetry.  The fractal generator, Apophysis, uses multiple julia-inspired variations (such as “julian”, “julia3D”, and “juliascope”) which can be utilized to manipulate existing fractals in a julia-like way.  The example image for this section was created using the aforementioned juliascope variation in Apophysis.  Any given point of a Mandelbrot set (see the spotlight on “Benoit Mandelbrot” in the Origins section) also generates its own unique julia set.

If a chaotic anomaly was a living creature with hopes and dreams, then hyperbolic geometry would be its Wonderland.  While even a chaotic anomaly such as this has specific mathematical rules, it is much easier to explain it in terms of its visual characteristics.  Imagine a simple, flat plane from regular Euclidean geometry (i.e. an infinite 2D surface).  The hyperbolic variation of that plane would be warped and bent and sunken and stretched into something of an abomination (the example image is an extreme version of this variation created in Apophysis).  Any geometry laid out on to this plane, such as lines, angles, and shapes, would be forced to warp to the same aberrance.  Circles are turned inside out.  The three angles of a triangle no longer equal 180 degrees.  Parallel lines twist and bend and undulate, yet still never cross each other.  Mice open their mouths and elephants jump out… okay, maybe not that.   Suffice it to say that objects affected by a hyperbolic variance inherit a slightly different, yet distinct set of mathematical rules.  The end result is often similar in appearance to the example image.  It appears as something of a complex web, a warped plane in space, or some form of multifaceted black hole.  The simplest form of hyperbolic plane resembles a plane that has been pushed outward from the center and tapers into a stretched cylindrical form in both directions.

A popcorn fractal is created using a complex formula involving sines and tangents.  While sines and tangents are not inherently chaotic, they have an interesting effect when applied within an anomalous function.  A sine produces wavy forms, and is the key mathematical concept in creating the visual representation of sound waves.  A tangent, on the other hand, is a line that touches the edge of a waveform or other non-linear curve, but does not cross it at that particular point.  When used together dynamically in a self-perpetuating formula, most often a julia set, the result is a design that curves, curls, and bounces off of itself, leaving popcorn-like holes in its geometry.  Many other unique effects can be created using sine, tangent, and even cosine, and so these functions are often considered to be variations in and of themselves.  The example image is a simple rendering of the default “popcorn julia” script in Incendia.

Mandelbrot setMandelbrot Set
Ah, the infamous Mandelbrot set, staple of graphic artists everywhere.  No list of variations would be even remotely complete without giving it a mention.  This script, developed by fractal pioneer, Benoit Mandelbrot (see the spotlight on “Benoit Mandelbrot” in the Origins section), is the product of a specific formula iteration.  A Mandelbrot set is formed using an equation very similar to that of a julia, and perpetuates itself exponentially.  The primary difference between a julia and a Mandelbrot set is that the result of the M-set is built along a complex plane.  The interior space of the plane is empty, while the exterior space is an expansion of formula iterations.  By drawing the fractal in this way, the iterations are built outward from an irregular surface, overlapping and interacting in unique patterns.  By zooming into the Mandelbrot set (to virtually infinite degrees), one can find an unending variety of unique fractal formations, making this variation one of the most versatile in its class right out of the box.  As this script is essentially a julia derivitave, any given point of a Mandelbrot set produces a unique julia set. This quality is utilized in most fractal programs containing this variation, allowing a designer to use the M-set as something of a julia hunting ground. The example image is the standard M-set formation, and was rendered in Ultra Fractal.